Physics 736. Experimental Methods in Nuclear-, Particle-, and Astrophysics. Lecture 14
|
|
- Moses Washington
- 5 years ago
- Views:
Transcription
1 Physics 736 Experimental Methods in Nuclear-, Particle-, and Astrophysics Lecture 14 Karsten Heeger
2 Course Schedule and Reading course website homework #5 due today homework #6 is due on April 5, 2010
3 Statistics &Numerical Techniques Topics confidence intervals and limits Bayesian and frequentist approach Feldman Cousing random numbers Monte Carlo Techniques
4 Confidence Intervals and Limits Karsten Heeger, Univ. of Wisconsin NUSS, July 13, 2009
5 Confidence Intervals and Limits Gaussian distribution
6 Confidence Intervals: Measurements and Limits rate or flux or # of events x confidence interval (CL=68.3%) confidence interval (CL=99%) What if some measurements are in a non-physical region? Does it matter whether we set a limit or a report a measurement?
7 Frequentist vs Bayesian Approach Two philosophies Bayesian approach probability = degree of belief that something will happen or that a parameter will have a given value Frequentist approach probability = relative frequency of something happening one can define frequentist probability for observing data (which are random) but not for the true value of a parameter independent of observer
8 Frequentist vs Bayesian Approach Two philosophies Bayesian approach require as input the prior beliefs of the physicist doing the analysis. necessarily subjective, and not allowed in frequentist method. Frequentist approach require as input probabilities of observing all data, including both the data actually observed and that which could have been observed (the Monte Carlo). not allowed in Bayesian method.
9 Frequentist vs Bayesian Approach classical approach Bayesian approach A precise experiment and an imprecise one with a statistical fluctuation can give the same limit! Q= prior belief function But it is not possible to combine the results of experiments that just quote a mass interval and confidence level.
10 Issue of Coverage Confidence intervals undercover Measurement pretends to be more accurate than it actually is Correct coverage Confidence intervals overcover (i.e. are too conservative) Reduced power to reject wrong hypotheses Proper coverage can be tested by Monte Carlo simulations
11 Flip-Flopping The flip-flopping attitude (example): We will state a measurement with a 1σ error (i.e. CL=68.3%) if the measurement result is above mσ, and an 99% CL upper limit otherwise. Flip-flopping between measurements and upper limits with different confidence levels spoils the coverage of the stated confidence intervals Easy to show with a toy Monte Carlo
12 Classical Confidence Intervals
13 Classical Confidence Intervals
14 Bayesian Interval
15 Example
16 Feldman & Cousins Approach Provides confidence intervals that change smoothly from upper limits to measurements User just needs to decide for a confidence level Flip-flopping problem is solved Uses Neyman s construction and a Likelihood Ratio to decide what values are included into confidence intervals
17 (Frequentist) Definition of the confidence interval for the measurement of a quantity x: If the experiment were repeated and in each attempt a confidence interval is calculated, then a fraction α of the confidence intervals will contain the true value of x (called µ). A fraction 1-α of the confidence intervals will not contain µ. Note: Experiments must not be identical
18 Random Number Monte Carlo Techniques Karsten Heeger, Univ. of Wisconsin NUSS, July 13, 2009
19 Random Numbers & Monte Carlo Techniques Who has used a Monte Carlo before? Who has written a Monte Carlo before? what are elements of Monte Carlo?
20 Random Numbers & Monte Carlo Techniques Monte Carlo (MC) refers to any procedure that makes use of random numbers MC methods are used in simulation of natural phenomena simulation of experimental apparatus numerical analysis (e.g. integration of many variables)
21 Random Numbers & Monte Carlo Techniques Simulating Data or Experiment
22 Random Numbers & Monte Carlo Techniques Simulating Radioactive Decay
23 Random Numbers & Monte Carlo Techniques Estimating the Area of a Circle ** * ** ++? 't-.r o o F o z -o x 2.s Area of circle hits from 100 pairs of random numbers uniformly distributed between -1 and +1 # of hits inside circle give area estimate circle area estimates obtained from 100 MC runs, each with 100 pairs of random numbers. Gaussian curve based on mean and standard deviation of 100 estimated areas.
24 Random Numbers & Monte Carlo Techniques Random Numbers What is a random number? Is 3 a random number?
25 Random Numbers & Monte Carlo Techniques Random Numbers What is a random number? Is 3 a random number? No such thing as a single random number. A sequence of random numbers or a set of numbers that have nothing to do with the other numbers in the sequence. In a uniform distribution of random numbers in the range of [0,1] every number has the sam chance of turning up is as likely as 0.5.
26 Random Numbers & Monte Carlo Techniques How to Generate Random Numbers chaotic system e.g. lottery random process radioactive decay thermal noise cosmic ray arrival random number tables computer code
27 Random Numbers & Monte Carlo Techniques Random Number Tables
28 Random Numbers & Monte Carlo Techniques How to Generate Random Numbers all algorithms produce a periodic sequence of numbers sequence of numbers in a uniform distribution in the range [0,1] algorithms generate integers between 0 and M and return a real value x n = I n /M to obtain effectively random values, use small subset of a single period e.g. Mersenne twister algorithm = long period
29 Random Numbers & Monte Carlo Techniques How to Generate Random Numbers Middle Square, Von Neumann, 1946 generate a sequence of 10 digit integers, start with one, square it, and then take the middle 10 digits from answer as next number in sequence sequence is not random since each number is completely determined from previous one. but it appears random. a more complex algorithm does not lead to a better random sequence. it is better to use an algorithm that is well understood.
30 Random Numbers & Monte Carlo Techniques RANDU from IBM in 1960s RANDU 2D I n+1 =(65539 I n )mod2 31 RANDU 3D
31 Random Numbers & Monte Carlo Techniques How to Generate Random Numbers not all random number generators are good! For example, in ROOT TRandom3 recommended by ROOT TRandom too short of a period For example, in Numerical Recipers authors have admitted that RAN1 and RAN2 in first edition are mediocre generators ran0, ran1, ran2 are much better in second edition
32 Random Numbers & Monte Carlo Techniques How to Improve Generators improve behavior and increase period my modifying algorithms I n =(a I n 1 + b I n 2 )mod m this has 2 initial seeds and can have a period greater than m RABMAR generator in CERNLIB requires 103 seeds. the ultimate random number generator.
33 Random Numbers & Monte Carlo Techniques Simulating Distributions so far we have only considered random number in [0,1] more complicated problems generally require random numbers generated according to specific distributions we can generate random numbers according to certain distributions (e.g Poisson for radioactive decay) can use special purpose algorithms. use numerical libraries and routines.
34 Acceptance/Rejection Method (von Neumann) Problem: generate a series of random numbers, xi, which follow a distribution f(x) Method: choose trial value, xtrial. accept with probability f(xtrial) choose trial x with random number λ1 x trial = x min +(x max x min )λ 1 random points are chosen inside the box and rejected if the ordinate exceeds f(x)
35 Acceptance/Rejection Method (von Neumann) random points are chosen inside the box and rejected if the ordinate exceeds f(x) bounding region is method to increase efficiency efficiency of method = ratio of areas keep Ch(x) as close as possible to f(x) Method applicable if - f(x) is too complex for other techniques - f(x) can be computed beware of normalization
36 Acceptance/Rejection Method (von Neumann) rejection algorithm is not efficient if the distribution has one or more larger peaks (or poles). in this case trial events are seldomly accepted. algorithm does not work when the range of x is [-, + ]
37 Inverse Transform Method applicable for simple distribution functions Method probability density function is f(x) in [-, + ] integrated probability up to point a is F(a) for x a F(a) is itself a random variable which will occur with uniform probability density on [0,1] we can find a unique x for a given u if u = F (x) provided we can find inverse x = F (u) 1
38 Inverse Transform Method Use of a random number u chosen from a uniform distribution [0,1] to find a random number x from a distribution with cumulative distribution function F(x) PDG
39 Inverse Transform Method Practical Method 1. normalize distribution function so that it becomes a probability distribution function (PDF) 2. integrate PDF from xmin to arbitrary x. this is probability of choosing a value less than x. 3. equate this to a uniform random number and solve for x. the resulting x will be distributed according to PDF. in other words, solve following equation for x given a uniform random number λ x x f(x)dx min xmax f(x)dx = λ x min
40 Inverse Transform Method convenient when you can calculate the inverse function e.g. exp(x), (1-x) n, 1/(1+x 2 ) there are som packages that do this for you. e.g. UNU.RAN in ROOT Examples generate x between 0 and 4 according to f(x) =x 0.5 generate x between 0 and according to f(x) =e x
41 Random Numbers & Monte Carlo Techniques What if rejection technique is impractical and you cannot invert the integral of the distribution function? Replace the distribution function f(x) by an approximate form f* (x)for which the inversion technique can be applied Generate trial values for x with inversion technique according to f*(x), and accept trial value with probability proportional to weight w = f(x)/f (x) f (x) rejection technique = special case where f*(x) is constant
42 Random Numbers & Monte Carlo Techniques Multidimensional Simulation (simulating a distribution in more than one dimension) if distribution is separable then variables are uncorrelated, each can be generated as before f(x, y) =g(x)h(y) generate x according to g(x) and y according to h(y) otherwise, distribution along each dimension needs to be calculated ymax D x (x) = f(x, y)dy y min f (x, y)dx find approximate distribution so that f (x, y)dy are invertible weights for trial events are given by w = f(x, y) f (x, y)
43 Monte Carlo Numbering Scheme To facilitate interfacing between event generators, detector simulators, and analysis packages used in particle physics
44 Karsten Heeger, Univ. of Wisconsin NUSS, July 13, 2009
Physics 736. Experimental Methods in Nuclear-, Particle-, and Astrophysics. - Statistics and Error Analysis -
Physics 736 Experimental Methods in Nuclear-, Particle-, and Astrophysics - Statistics and Error Analysis - Karsten Heeger heeger@wisc.edu Feldman&Cousin what are the issues they deal with? what processes
More informationPhysics 736. Experimental Methods in Nuclear-, Particle-, and Astrophysics. - Statistical Methods -
Physics 736 Experimental Methods in Nuclear-, Particle-, and Astrophysics - Statistical Methods - Karsten Heeger heeger@wisc.edu Course Schedule and Reading course website http://neutrino.physics.wisc.edu/teaching/phys736/
More informationComputational Methods. Randomness and Monte Carlo Methods
Computational Methods Randomness and Monte Carlo Methods Manfred Huber 2010 1 Randomness and Monte Carlo Methods Introducing randomness in an algorithm can lead to improved efficiencies Random sampling
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 13 Random Numbers and Stochastic Simulation Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright
More informationMonte Carlo Integration and Random Numbers
Monte Carlo Integration and Random Numbers Higher dimensional integration u Simpson rule with M evaluations in u one dimension the error is order M -4! u d dimensions the error is order M -4/d u In general
More informationMULTI-DIMENSIONAL MONTE CARLO INTEGRATION
CS580: Computer Graphics KAIST School of Computing Chapter 3 MULTI-DIMENSIONAL MONTE CARLO INTEGRATION 2 1 Monte Carlo Integration This describes a simple technique for the numerical evaluation of integrals
More informationWhat is the Monte Carlo Method?
Program What is the Monte Carlo Method? A bit of history Applications The core of Monte Carlo: Random realizations 1st example: Initial conditions for N-body simulations 2nd example: Simulating a proper
More informationNumerical Integration
Lecture 12: Numerical Integration (with a focus on Monte Carlo integration) Computer Graphics CMU 15-462/15-662, Fall 2015 Review: fundamental theorem of calculus Z b f(x)dx = F (b) F (a) a f(x) = d dx
More informationSampling and Monte-Carlo Integration
Sampling and Monte-Carlo Integration Sampling and Monte-Carlo Integration Last Time Pixels are samples Sampling theorem Convolution & multiplication Aliasing: spectrum replication Ideal filter And its
More informationLecture 2: Introduction to Numerical Simulation
Lecture 2: Introduction to Numerical Simulation Ahmed Kebaier kebaier@math.univ-paris13.fr HEC, Paris Outline of The Talk 1 Simulation of Random variables Outline 1 Simulation of Random variables Random
More information1. Practice the use of the C ++ repetition constructs of for, while, and do-while. 2. Use computer-generated random numbers.
1 Purpose This lab illustrates the use of looping structures by introducing a class of programming problems called numerical algorithms. 1. Practice the use of the C ++ repetition constructs of for, while,
More informationVARIANCE REDUCTION TECHNIQUES IN MONTE CARLO SIMULATIONS K. Ming Leung
POLYTECHNIC UNIVERSITY Department of Computer and Information Science VARIANCE REDUCTION TECHNIQUES IN MONTE CARLO SIMULATIONS K. Ming Leung Abstract: Techniques for reducing the variance in Monte Carlo
More informationChapter 6: Simulation Using Spread-Sheets (Excel)
Chapter 6: Simulation Using Spread-Sheets (Excel) Refer to Reading Assignments 1 Simulation Using Spread-Sheets (Excel) OBJECTIVES To be able to Generate random numbers within a spreadsheet environment.
More informationBootstrapping Method for 14 June 2016 R. Russell Rhinehart. Bootstrapping
Bootstrapping Method for www.r3eda.com 14 June 2016 R. Russell Rhinehart Bootstrapping This is extracted from the book, Nonlinear Regression Modeling for Engineering Applications: Modeling, Model Validation,
More informationIntroduction to Mobile Robotics Bayes Filter Particle Filter and Monte Carlo Localization. Wolfram Burgard
Introduction to Mobile Robotics Bayes Filter Particle Filter and Monte Carlo Localization Wolfram Burgard 1 Motivation Recall: Discrete filter Discretize the continuous state space High memory complexity
More informationRandom Numbers and Monte Carlo Methods
Random Numbers and Monte Carlo Methods Methods which make use of random numbers are often called Monte Carlo Methods after the Monte Carlo Casino in Monaco which has long been famous for games of chance.
More informationYou ve already read basics of simulation now I will be taking up method of simulation, that is Random Number Generation
Unit 5 SIMULATION THEORY Lesson 39 Learning objective: To learn random number generation. Methods of simulation. Monte Carlo method of simulation You ve already read basics of simulation now I will be
More informationProbabilistic Robotics
Probabilistic Robotics Discrete Filters and Particle Filters Models Some slides adopted from: Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Kai Arras and Probabilistic Robotics Book SA-1 Probabilistic
More informationChapter 1. Introduction
Chapter 1 Introduction A Monte Carlo method is a compuational method that uses random numbers to compute (estimate) some quantity of interest. Very often the quantity we want to compute is the mean of
More informationDealing with Categorical Data Types in a Designed Experiment
Dealing with Categorical Data Types in a Designed Experiment Part II: Sizing a Designed Experiment When Using a Binary Response Best Practice Authored by: Francisco Ortiz, PhD STAT T&E COE The goal of
More informationMonte Carlo Integration
Lecture 11: Monte Carlo Integration Computer Graphics and Imaging UC Berkeley CS184/284A, Spring 2016 Reminder: Quadrature-Based Numerical Integration f(x) Z b a f(x)dx x 0 = a x 1 x 2 x 3 x 4 = b E.g.
More informationSimulation. Monte Carlo
Simulation Monte Carlo Monte Carlo simulation Outcome of a single stochastic simulation run is always random A single instance of a random variable Goal of a simulation experiment is to get knowledge about
More informationMETROPOLIS MONTE CARLO SIMULATION
POLYTECHNIC UNIVERSITY Department of Computer and Information Science METROPOLIS MONTE CARLO SIMULATION K. Ming Leung Abstract: The Metropolis method is another method of generating random deviates that
More informationQuantitative Biology II!
Quantitative Biology II! Lecture 3: Markov Chain Monte Carlo! March 9, 2015! 2! Plan for Today!! Introduction to Sampling!! Introduction to MCMC!! Metropolis Algorithm!! Metropolis-Hastings Algorithm!!
More informationLecture 7: Monte Carlo Rendering. MC Advantages
Lecture 7: Monte Carlo Rendering CS 6620, Spring 2009 Kavita Bala Computer Science Cornell University MC Advantages Convergence rate of O( ) Simple Sampling Point evaluation Can use black boxes General
More informationMonte Carlo Methods and Statistical Computing: My Personal E
Monte Carlo Methods and Statistical Computing: My Personal Experience Department of Mathematics & Statistics Indian Institute of Technology Kanpur November 29, 2014 Outline Preface 1 Preface 2 3 4 5 6
More informationProbability and Statistics for Final Year Engineering Students
Probability and Statistics for Final Year Engineering Students By Yoni Nazarathy, Last Updated: April 11, 2011. Lecture 1: Introduction and Basic Terms Welcome to the course, time table, assessment, etc..
More informationISyE 6416: Computational Statistics Spring Lecture 13: Monte Carlo Methods
ISyE 6416: Computational Statistics Spring 2017 Lecture 13: Monte Carlo Methods Prof. Yao Xie H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology Determine area
More information>>> SOLUTIONS <<< a) What is a system? Give a short formal definition. (Hint: what you draw a box around is not a formal definition)
Mid-Term Exam for Simulation (CIS 4930) Summer 2009 >>> SOLUTIONS
More informationBiostatistics 615/815 Lecture 16: Importance sampling Single dimensional optimization
Biostatistics 615/815 Lecture 16: Single dimensional optimization Hyun Min Kang November 1st, 2012 Hyun Min Kang Biostatistics 615/815 - Lecture 16 November 1st, 2012 1 / 59 The crude Monte-Carlo Methods
More informationComputer Vision Group Prof. Daniel Cremers. 11. Sampling Methods
Prof. Daniel Cremers 11. Sampling Methods Sampling Methods Sampling Methods are widely used in Computer Science as an approximation of a deterministic algorithm to represent uncertainty without a parametric
More informationLearning Objectives. Continuous Random Variables & The Normal Probability Distribution. Continuous Random Variable
Learning Objectives Continuous Random Variables & The Normal Probability Distribution 1. Understand characteristics about continuous random variables and probability distributions 2. Understand the uniform
More informationMonte Carlo Techniques. Professor Stephen Sekula Guest Lecture PHY 4321/7305 Sep. 3, 2014
Monte Carlo Techniques Professor Stephen Sekula Guest Lecture PHY 431/7305 Sep. 3, 014 What are Monte Carlo Techniques? Computational algorithms that rely on repeated random sampling in order to obtain
More informationRandom Numbers Random Walk
Random Numbers Random Walk Computational Physics Random Numbers Random Walk Outline Random Systems Random Numbers Monte Carlo Integration Example Random Walk Exercise 7 Introduction Random Systems Deterministic
More informationMotivation. Advanced Computer Graphics (Fall 2009) CS 283, Lecture 11: Monte Carlo Integration Ravi Ramamoorthi
Advanced Computer Graphics (Fall 2009) CS 283, Lecture 11: Monte Carlo Integration Ravi Ramamoorthi http://inst.eecs.berkeley.edu/~cs283 Acknowledgements and many slides courtesy: Thomas Funkhouser, Szymon
More informationThe Plan: Basic statistics: Random and pseudorandom numbers and their generation: Chapter 16.
Scientific Computing with Case Studies SIAM Press, 29 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit IV Monte Carlo Computations Dianne P. O Leary c 28 What is a Monte-Carlo method?
More informationProbabilistic (Randomized) algorithms
Probabilistic (Randomized) algorithms Idea: Build algorithms using a random element so as gain improved performance. For some cases, improved performance is very dramatic, moving from intractable to tractable.
More informationACCURACY AND EFFICIENCY OF MONTE CARLO METHOD. Julius Goodman. Bechtel Power Corporation E. Imperial Hwy. Norwalk, CA 90650, U.S.A.
- 430 - ACCURACY AND EFFICIENCY OF MONTE CARLO METHOD Julius Goodman Bechtel Power Corporation 12400 E. Imperial Hwy. Norwalk, CA 90650, U.S.A. ABSTRACT The accuracy of Monte Carlo method of simulating
More informationScientific Computing with Case Studies SIAM Press, Lecture Notes for Unit IV Monte Carlo
Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit IV Monte Carlo Computations Dianne P. O Leary c 2008 1 What is a Monte-Carlo
More informationIntroduction to hypothesis testing
Introduction to hypothesis testing Mark Johnson Macquarie University Sydney, Australia February 27, 2017 1 / 38 Outline Introduction Hypothesis tests and confidence intervals Classical hypothesis tests
More informationMonte Carlo Integration COS 323
Monte Carlo Integration COS 323 Last time Interpolatory Quadrature Review formulation; error analysis Newton-Cotes Quadrature Midpoint, Trapezoid, Simpson s Rule Error analysis for trapezoid, midpoint
More informationDesign of Experiments
Seite 1 von 1 Design of Experiments Module Overview In this module, you learn how to create design matrices, screen factors, and perform regression analysis and Monte Carlo simulation using Mathcad. Objectives
More informationA noninformative Bayesian approach to small area estimation
A noninformative Bayesian approach to small area estimation Glen Meeden School of Statistics University of Minnesota Minneapolis, MN 55455 glen@stat.umn.edu September 2001 Revised May 2002 Research supported
More informationRANDOM NUMBERS GENERATION
Chapter 4 RANDOM NUMBERS GENERATION M. Ragheb 10/2/2015 4.1. INTRODUCTION The use and generation of random numbers uniformly distributed over the unit interval: [0, 1] is a unique feature of the Monte
More informationMetropolis Light Transport
Metropolis Light Transport CS295, Spring 2017 Shuang Zhao Computer Science Department University of California, Irvine CS295, Spring 2017 Shuang Zhao 1 Announcements Final presentation June 13 (Tuesday)
More informationRandom Number Generation. Biostatistics 615/815 Lecture 16
Random Number Generation Biostatistics 615/815 Lecture 16 Some Uses of Random Numbers Simulating data Evaluate statistical procedures Evaluate study designs Evaluate program implementations Controlling
More informationSTAT 725 Notes Monte Carlo Integration
STAT 725 Notes Monte Carlo Integration Two major classes of numerical problems arise in statistical inference: optimization and integration. We have already spent some time discussing different optimization
More informationGAMES Webinar: Rendering Tutorial 2. Monte Carlo Methods. Shuang Zhao
GAMES Webinar: Rendering Tutorial 2 Monte Carlo Methods Shuang Zhao Assistant Professor Computer Science Department University of California, Irvine GAMES Webinar Shuang Zhao 1 Outline 1. Monte Carlo integration
More informationMonte Carlo Ray Tracing. Computer Graphics CMU /15-662
Monte Carlo Ray Tracing Computer Graphics CMU 15-462/15-662 TODAY: Monte Carlo Ray Tracing How do we render a photorealistic image? Put together many of the ideas we ve studied: - color - materials - radiometry
More informationMonte Carlo Integration
Lecture 15: Monte Carlo Integration Computer Graphics and Imaging UC Berkeley Reminder: Quadrature-Based Numerical Integration f(x) Z b a f(x)dx x 0 = a x 1 x 2 x 3 x 4 = b E.g. trapezoidal rule - estimate
More informationDiscussion on Bayesian Model Selection and Parameter Estimation in Extragalactic Astronomy by Martin Weinberg
Discussion on Bayesian Model Selection and Parameter Estimation in Extragalactic Astronomy by Martin Weinberg Phil Gregory Physics and Astronomy Univ. of British Columbia Introduction Martin Weinberg reported
More informationSection 1.6. Inverse Functions
Section 1.6 Inverse Functions Important Vocabulary Inverse function: Let f and g be two functions. If f(g(x)) = x in the domain of g and g(f(x) = x for every x in the domain of f, then g is the inverse
More informationRANDOM NUMBERS GENERATION
Chapter 4 RANDOM NUMBERS GENERATION M. Ragheb 9//2013 4.1. INTRODUCTION The use and generation of random numbers uniformly distributed over the unit interval: [0, 1] is a unique feature of the Monte Carlo
More informationWhat We ll Do... Random
What We ll Do... Random- number generation Random Number Generation Generating random variates Nonstationary Poisson processes Variance reduction Sequential sampling Designing and executing simulation
More informationScience Textbook and Instructional Materials Correlation to the 2010 Physics Standards of Learning and Curriculum Framework. Publisher Information
Publisher Information Copyright date 2009 Contact Carol Kornfeind Phone# 847-486-2065 E-mail carol.kornfeind@pearson.com Physics 1 of 16 Virginia Department of Education PH.1 The student will plan and
More informationMonte Carlo and Numerical Methods
Monte Carlo and Numerical Methods Scott Oser Lecture #4 Physics 509 1 Outline Last time: we studied Poisson, exponential, and 2 distributions, and learned how to generate new PDFs from other PDFs by marginalizing,
More informationGT "Calcul Ensembliste"
GT "Calcul Ensembliste" Beyond the bounded error framework for non linear state estimation Fahed Abdallah Université de Technologie de Compiègne 9 Décembre 2010 Fahed Abdallah GT "Calcul Ensembliste" 9
More informationProfessor Stephen Sekula
Monte Carlo Techniques Professor Stephen Sekula Guest Lecture PHYS 4321/7305 What are Monte Carlo Techniques? Computational algorithms that rely on repeated random sampling in order to obtain numerical
More informationA simple OMNeT++ queuing experiment using different random number generators
A simple OMNeT++ queuing experiment using different random number generators Bernhard Hechenleitner and Karl Entacher December 5, 2002 Abstract We apply a simple queuing-experiment using parallel streams
More informationCosmic Ray Shower Profile Track Finding for Telescope Array Fluorescence Detectors
Cosmic Ray Shower Profile Track Finding for Telescope Array Fluorescence Detectors High Energy Astrophysics Institute and Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah,
More informationCS 563 Advanced Topics in Computer Graphics Monte Carlo Integration: Basic Concepts. by Emmanuel Agu
CS 563 Advanced Topics in Computer Graphics Monte Carlo Integration: Basic Concepts by Emmanuel Agu Introduction The integral equations generally don t have analytic solutions, so we must turn to numerical
More information10.4 Linear interpolation method Newton s method
10.4 Linear interpolation method The next best thing one can do is the linear interpolation method, also known as the double false position method. This method works similarly to the bisection method by
More informationConvexization in Markov Chain Monte Carlo
in Markov Chain Monte Carlo 1 IBM T. J. Watson Yorktown Heights, NY 2 Department of Aerospace Engineering Technion, Israel August 23, 2011 Problem Statement MCMC processes in general are governed by non
More information14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationToday s outline: pp
Chapter 3 sections We will SKIP a number of sections Random variables and discrete distributions Continuous distributions The cumulative distribution function Bivariate distributions Marginal distributions
More informationChapter 7: Computation of the Camera Matrix P
Chapter 7: Computation of the Camera Matrix P Arco Nederveen Eagle Vision March 18, 2008 Arco Nederveen (Eagle Vision) The Camera Matrix P March 18, 2008 1 / 25 1 Chapter 7: Computation of the camera Matrix
More informationStarting a Data Analysis
03/20/07 PHY310: Statistical Data Analysis 1 PHY310: Lecture 17 Starting a Data Analysis Road Map Your Analysis Log Exploring the Data Reading the input file (and making sure it's right) Taking a first
More informationAn Introduction to Markov Chain Monte Carlo
An Introduction to Markov Chain Monte Carlo Markov Chain Monte Carlo (MCMC) refers to a suite of processes for simulating a posterior distribution based on a random (ie. monte carlo) process. In other
More informationMotivation. Monte Carlo Path Tracing. Monte Carlo Path Tracing. Monte Carlo Path Tracing. Monte Carlo Path Tracing
Advanced Computer Graphics (Spring 2013) CS 283, Lecture 11: Monte Carlo Path Tracing Ravi Ramamoorthi http://inst.eecs.berkeley.edu/~cs283/sp13 Motivation General solution to rendering and global illumination
More informationReproducibility in Stochastic Simulation
Reproducibility in Stochastic Simulation Prof. Michael Mascagni Department of Computer Science Department of Mathematics Department of Scientific Computing Graduate Program in Molecular Biophysics Florida
More informationOptimal designs for comparing curves
Optimal designs for comparing curves Holger Dette, Ruhr-Universität Bochum Maria Konstantinou, Ruhr-Universität Bochum Kirsten Schorning, Ruhr-Universität Bochum FP7 HEALTH 2013-602552 Outline 1 Motivation
More informationPage 129 Exercise 5: Suppose that the joint p.d.f. of two random variables X and Y is as follows: { c(x. 0 otherwise. ( 1 = c. = c
Stat Solutions for Homework Set Page 9 Exercise : Suppose that the joint p.d.f. of two random variables X and Y is as follows: { cx fx, y + y for y x, < x < otherwise. Determine a the value of the constant
More informationFUNCTIONS AND MODELS
1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS In this section, we assume that you have access to a graphing calculator or a computer with graphing software. FUNCTIONS AND MODELS 1.4 Graphing Calculators
More information13 Distribution Ray Tracing
13 In (hereafter abbreviated as DRT ), our goal is to render a scene as accurately as possible. Whereas Basic Ray Tracing computed a very crude approximation to radiance at a point, in DRT we will attempt
More informationLecture 8: Jointly distributed random variables
Lecture : Jointly distributed random variables Random Vectors and Joint Probability Distributions Definition: Random Vector. An n-dimensional random vector, denoted as Z = (Z, Z,, Z n ), is a function
More informationAUTONOMOUS SYSTEMS. PROBABILISTIC LOCALIZATION Monte Carlo Localization
AUTONOMOUS SYSTEMS PROBABILISTIC LOCALIZATION Monte Carlo Localization Maria Isabel Ribeiro Pedro Lima With revisions introduced by Rodrigo Ventura Instituto Superior Técnico/Instituto de Sistemas e Robótica
More informationOverview. Monte Carlo Methods. Statistics & Bayesian Inference Lecture 3. Situation At End Of Last Week
Statistics & Bayesian Inference Lecture 3 Joe Zuntz Overview Overview & Motivation Metropolis Hastings Monte Carlo Methods Importance sampling Direct sampling Gibbs sampling Monte-Carlo Markov Chains Emcee
More informationWill Monroe July 21, with materials by Mehran Sahami and Chris Piech. Joint Distributions
Will Monroe July 1, 017 with materials by Mehran Sahami and Chris Piech Joint Distributions Review: Normal random variable An normal (= Gaussian) random variable is a good approximation to many other distributions.
More informationMonte Carlo Integration COS 323
Monte Carlo Integration COS 323 Integration in d Dimensions? One option: nested 1-D integration f(x,y) g(y) y f ( x, y) dx dy ( ) = g y dy x Evaluate the latter numerically, but each sample of g(y) is
More informationMonte Carlo for Spatial Models
Monte Carlo for Spatial Models Murali Haran Department of Statistics Penn State University Penn State Computational Science Lectures April 2007 Spatial Models Lots of scientific questions involve analyzing
More informationBESTFIT, DISTRIBUTION FITTING SOFTWARE BY PALISADE CORPORATION
Proceedings of the 1996 Winter Simulation Conference ed. J. M. Charnes, D. J. Morrice, D. T. Brunner, and J. J. S\vain BESTFIT, DISTRIBUTION FITTING SOFTWARE BY PALISADE CORPORATION Linda lankauskas Sam
More informationCategorical Data in a Designed Experiment Part 2: Sizing with a Binary Response
Categorical Data in a Designed Experiment Part 2: Sizing with a Binary Response Authored by: Francisco Ortiz, PhD Version 2: 19 July 2018 Revised 18 October 2018 The goal of the STAT COE is to assist in
More informationComputational Physics Adaptive, Multi-Dimensional, & Monte Carlo Integration Feb 21, 2019
Computational Physics Adaptive, Multi-Dimensional, & Monte Carlo Integration Feb 21, 219 http://hadron.physics.fsu.edu/~eugenio/comphy/ eugenio@fsu.edu f(x) trapezoidal rule Series Integration review from
More informationTutorial: Random Number Generation
Tutorial: Random Number Generation John Lau johnlau@umail.ucsb.edu Henry Yu henryyu@umail.ucsb.edu June 2018 1 Abstract This tutorial will cover the basics of Random Number Generation. This includes properties
More informationAn interesting related problem is Buffon s Needle which was first proposed in the mid-1700 s.
Using Monte Carlo to Estimate π using Buffon s Needle Problem An interesting related problem is Buffon s Needle which was first proposed in the mid-1700 s. Here s the problem (in a simplified form). Suppose
More informationPhoton Maps. The photon map stores the lighting information on points or photons in 3D space ( on /near 2D surfaces)
Photon Mapping 1/36 Photon Maps The photon map stores the lighting information on points or photons in 3D space ( on /near 2D surfaces) As opposed to the radiosity method that stores information on surface
More information10.2 Applications of Monte Carlo Methods
Chapter 10 Monte Carlo Methods There is no such thing as a perfectly random number. teacher - Harold Bailey, my 8 th grade math Preface When I was a youngster, I was the captain of my junior high school
More informationStatistical techniques for data analysis in Cosmology
Statistical techniques for data analysis in Cosmology arxiv:0712.3028; arxiv:0911.3105 Numerical recipes (the bible ) Licia Verde ICREA & ICC UB-IEEC http://icc.ub.edu/~liciaverde outline Lecture 1: Introduction
More informationPh3 Mathematica Homework: Week 8
Ph3 Mathematica Homework: Week 8 Eric D. Black California Institute of Technology v1.1 Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has
More informationNested Sampling: Introduction and Implementation
UNIVERSITY OF TEXAS AT SAN ANTONIO Nested Sampling: Introduction and Implementation Liang Jing May 2009 1 1 ABSTRACT Nested Sampling is a new technique to calculate the evidence, Z = P(D M) = p(d θ, M)p(θ
More informationSemantic Importance Sampling for Statistical Model Checking
Semantic Importance Sampling for Statistical Model Checking Software Engineering Institute Carnegie Mellon University Pittsburgh, PA 15213 Jeffery Hansen, Lutz Wrage, Sagar Chaki, Dionisio de Niz, Mark
More informationMath 494: Mathematical Statistics
Math 494: Mathematical Statistics Instructor: Jimin Ding jmding@wustl.edu Department of Mathematics Washington University in St. Louis Class materials are available on course website (www.math.wustl.edu/
More informationLecture 4. Digital Image Enhancement. 1. Principle of image enhancement 2. Spatial domain transformation. Histogram processing
Lecture 4 Digital Image Enhancement 1. Principle of image enhancement 2. Spatial domain transformation Basic intensity it tranfomation ti Histogram processing Principle Objective of Enhancement Image enhancement
More informationOff-Line and Real-Time Methods for ML-PDA Target Validation
Off-Line and Real-Time Methods for ML-PDA Target Validation Wayne R. Blanding*, Member, IEEE, Peter K. Willett, Fellow, IEEE and Yaakov Bar-Shalom, Fellow, IEEE 1 Abstract We present two procedures for
More informationHierarchical Bayesian Modeling with Ensemble MCMC. Eric B. Ford (Penn State) Bayesian Computing for Astronomical Data Analysis June 12, 2014
Hierarchical Bayesian Modeling with Ensemble MCMC Eric B. Ford (Penn State) Bayesian Computing for Astronomical Data Analysis June 12, 2014 Simple Markov Chain Monte Carlo Initialise chain with θ 0 (initial
More informationGenerating random samples from user-defined distributions
The Stata Journal (2011) 11, Number 2, pp. 299 304 Generating random samples from user-defined distributions Katarína Lukácsy Central European University Budapest, Hungary lukacsy katarina@phd.ceu.hu Abstract.
More informationPrime Time (Factors and Multiples)
CONFIDENCE LEVEL: Prime Time Knowledge Map for 6 th Grade Math Prime Time (Factors and Multiples). A factor is a whole numbers that is multiplied by another whole number to get a product. (Ex: x 5 = ;
More informationComputer Vision 2 Lecture 8
Computer Vision 2 Lecture 8 Multi-Object Tracking (30.05.2016) leibe@vision.rwth-aachen.de, stueckler@vision.rwth-aachen.de RWTH Aachen University, Computer Vision Group http://www.vision.rwth-aachen.de
More informationThe Normal Distribution
The Normal Distribution Lecture 20 Section 6.3.1 Robb T. Koether Hampden-Sydney College Wed, Sep 28, 2011 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 1 / 41 Outline
More informationPair-Wise Multiple Comparisons (Simulation)
Chapter 580 Pair-Wise Multiple Comparisons (Simulation) Introduction This procedure uses simulation analyze the power and significance level of three pair-wise multiple-comparison procedures: Tukey-Kramer,
More information